We consider signals defined on a real-world graph, given by the Euroroad network [1]}, obtained from the repository http://networkrepository.com/inf-euroroad.php. The Euroraod network is a real road network connecting various cities in Europe, having a total of around 1200 vertices and about 1400 edges. We first extract the largest connected component of the network, which results in 1039 vertices and 1305 edges. Next, to generate true piecewise constant signals on this network, we extract non-overlapping communities from the network using the Louvain community detection algorithm [2]}. The Louvain algorithm is one of the most popular community detection algorithms available in the literature which detects underlying communities in a network via optimising the modularity of the same. We use the cluster_louvain function available in the igraph package in R to extract the communities. The method results in 25 connected communities, with the smallest community having 11 vertices and the largest community having 82 vertices as members. The true signal on vertex \(j\)  with community label \(k\)  is given by \(\theta _{0,j} = k 1\!\mathrm {l}\lbrace \mathrm {vertex}\, j \in \mathrm {community}\, k\rbrace ,\, 1 \le j \le n.\)  Finally we generate a noisy version of the signal on the network given by \(y_j \sim N(\theta _{0,j}, 0.3^2),\, 1 \le j \le n.\)
